RADIATION HEAT TRANSFER

Heat conduction and convection - always a fluid which transfers the heat (gas, liquid, solid) – motion of atoms or molecules Heat conduction and convection is not possible in a vacuum In most practical applications all three modes occur concurrently at varying degrees

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A hot object in a vacuum chamber looses heat by radiation only n tio c e nv o c

Unlike conduction and convection, heat transfer by radiation can occur between two bodies, even when they are separated by a medium colder than both of them

at di ra n io

What will be a final equilibrium temperature of the body surface? Can you write an energy balance equation between the body and 2 surrounding air and the hot source (fire)?

Theoretical foundation of radiation was established by Maxwell Electromagnetic wave motion or electromagnetic radiation Electromagnetic waves travel at the speed of light c in a vacuum Electromagnetic waves are characterized by their frequency f or wavelength λ: λ=c/f co light speed in a vacuum c=co/n n refraction index of a medium (n=1 for air and most gases, n=1,5 for glass, 1,33 for water) In all material medium, there is attenuation of the energy In a vacuum there is no attenuation of the energy 3

Electromagnetic radiation covers a wide range of wavelengths Radiation that is related to heat transfer – Thermal radiation λ from 0,1μm to 100 μm As a result of energy transition in molecules, atoms and electrons. Thermal radiation is emitted by all matter whose temperature is above absolute zero. Everything around us emits (and absorbs) radiation. 4

• Thermal radiation includes entire visible (0,4 to 0,76 μm) and infrared light and a portion of ultraviolet radiation. • Body that emits radiation in the visible range is called light source. • Sun (primary light source) emits solar radiation – 0,3 to 3 μm – almost half is visible, remaining is ultraviolet and infrared. • Bodies at room temperature emit radiation in infrared range 0,7 to 100 μm. • Bodies start emit visible radiation at 800K (red hot) and tungsten wire in the lightbulb at 2000K (white hot) to emit a significant amount of radiation in the visible range.

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Spectral and Directional Distribution Radiation characteristics vary with wavelength and direction • Monochromatic or spectral: Characteristics at a given λ • Total: Integrated values over all wavelengths • Directional: At a given direction • Hemispherical: Integrated values over all directions • Diffuse radiation: Uniform in all directions The assumption of diffuse radiation will be made throughout

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Emissive Power E, Irradiation G and Radiosity J Eλ

• Emissive Power (zářivost): Radiation emitted from a surface • Spectral emissive power E λ :

E=

∞

∫0 Eλ dλ λ

Fig. 10.1

E λ = rate of emitted radiation per unit area per unit wavelength, W/m 2 μm • Total emissive power E: 2 λ W/m E = Integration of E λ over all values of , :

∞

E (T ) = E λ (λ , T ) dλ

∫0

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• Irradiation: Radiation energy incident on a surface • Spectral irradiation Gλ : Gλ = rate of radiation energy incident upon a surface 2 − μm W/m per unit area per unit wavelength, • Total irradiation G: G = integration of G λ over all values of λ : ∞

G (T ) = Gλ (λ , T ) dλ

∫0

• Radiosity: The sum of emitted and reflected radiation • Spectral radiosity J λ : J λ = rate of radiation leaving a surface per unit area per unit wavelength, W/m 2− μm 8

• Total radiosity J : J = integration of J λ over all values of λ : ∞

J (T ) = J λ (λ , T ) dλ

∫0

In the above definitions, summation in all directions is implied although the term hemispherical is not used

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Blackbody Radiation Blackbody:

An ideal radiation surface used as standard for describing radiation of real surfaces

Characteristics of blackbody: (1) It absorbs all radiation incident upon it (2) It emits the maximum energy at a given temperature and wavelength (3) Its emission is diffuse

Planck's Law E bλ = spectral emissive power of a blackbody: C 1λ− 5 E bλ (λ , T ) = C1 and C2 are constants exp(C 2 / λT ) − 1 10

Blackbody Radiation Maximum emitted energy at specific temperatures given by Wien law:

λmaxT = 2879,6 Thermal radiation 0,1 to 100 μm

Planck's Law

Note - by qualitative judgment energy emitted in visible range for 2000 K – tungsten wire in a light bulb. 11

Stefan-Boltzmann Law Based on: • Experimental data by Stefan (1879) • Theoretical derivation by Boltzmann (1884)

Eb = σ T 4

Stefan-Boltzmann law

E b = total blackbody emissive power (all wavelengths and all directions), [W/m2]

σ = 5.67 × 10 - 8 W/m 2− K 4 is the Stefan-Boltzmann constant It can also be arrived at using Planck's law 12

∞

Eb (T ) = ∫ Ebλ ( λ,T ) dx = 0

C 1 λ −5 =∫ dλ = 0 exp (C /λ T) − 1 2 ∞

= σT 4 • Stefan-Boltzmann law gives the total radiation emitted from a black body at all wavelengths from λ=0 to λ=∞. • Often an interest in radiation over some wavelength band – light bulb – how much is emitted in the visible range? • We use a procedure to determine Eb,0-λ λ

Eb,0 − λ(T) = ∫ Ebλ(λ , T)dλ 0

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Define a dimensionless quantity fλ(T):

f λ(T) =

λ ∫0 Eb,λ(T)dλ 4

σT

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Light bulb. Want to know how energy is emitted in the visible range 0,40 to 0,76 μm. λ1T=0,40.2500=1000 ⇒ fλ1 = 0,000321 λ2T=0, 76.2500=1900 ⇒ fλ2 = 0,053035

fλ2 - fλ1 = 0,0527 Only about 5% of radiation is emitted in the visible range. The remaining 95% is in the infrared region in the form of heat.

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Radiation of Real Surfaces Objective: Develop a methodology for determining radiation heat exchange between real surfaces. • Surface radiation properties • The graybody • Kirchhoff's law

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Absorptivity a, Reflectivity r, Transmissivity t J

G

rG ρG

E

αG

tG τG Fig. 10.2

Irradiation incident on a real surface can be absorbed, reflected and transmitted. Remind: radiosity J (total radiation leaving the surface) is a sum of emitted E and reflected rG radiation.

a = total absorptivity = fraction absorbed r = total reflectivity = fraction reflected t = total transmissivity = fraction transmitted aG + rG + tG = G a + r + t =1 17

Similarly

aλ + rλ + tλ = 1 aλ = spectral absorptivity rλ = spectral reflectivity tλ = spectral transmissivity

Opaque material: Simplification:

t = tλ = 0 a + r =1

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Emissivity (emisivita, poměrná zářivost) Total emissivity ε(T): Ratio of emissive power of a E λ surface to that of a blackbody at the same temperature:

E (T ) ε (T ) = Eb (T )

blackbody

real surface

λ

Fig. 10.3

Spectral emissivity ε λ : Ratio of the spectral emissive power of a surface to that of a blackbody at the same temperature: Eλ (λ , T ) ε λ (λ , T ) = Ebλ (λ , T ) 19

Kirchhoff's Law It is much easier to determine emissivity ε than absorptivity a. By experiments. But how we can determine absorptivity? Kirchhoff’s law says that under certain conditions: Spectral ε λ (λ , T ) = α λ (λ , T ) Total ε (T ) = α (T ) Kirchhoff’s law is used to determine aλ(λ,T) from experimental data on ελ(λ,T) Equality of emissivity and absorptivity Quite different physical quantities Just numerical equality

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Graybody Approximation The graybody concept is introduced to simplify the analysis of radiation exchange between bodies Graybody: An ideal surface for which the spectral emissivity ελ is independent of λ Eλ

blackbody

real surface gray body Fig. 10.3

Eλ

λ

approx. 0,75 Eb

blackbody

gray body

Fig. 10.3

λ

21

Thus:

ε λ (λ , T ) = ε (T ) = constant independent of λ

It follows from Kirchhoff's Law that

ε (T ) = α (T ) for a graybody

NOTE: (1) Radiation properties ε, a and r are assigned single values instead of a spectrum of values

(2) Data on ε give r and a for opaque surface. 22

Radiation Exchange Between Black Surfaces Two black surfaces with areas S1 and S 2 at temperatures T1 and T2

T1 > T2

S2 Q& 12

T1 1

S1 E 1

Objective: Determine the net heat transfer Q& 1− 2 between the two surfaces Important factors: • Configuration • Surface area • Surface temperature • Radiation properties (for gray body) • Surrounding surfaces • Space medium

2

T2 E2

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The View Factor (1) Definition and use: The view factor is the fraction of radiation energy leaving surface S1 which is intercepted by S2

• It is a geometric factor • Also known as shape factor and configuration factor

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S2 Q& 1− 2 T1 1

S1 E 1

2

T2 E2

Q& 1 = rate of radiation energy leaving surface 1, = S1E1 Q& 2 = rate of radiation energy leaving surface 2, = S2E2 Q& 1− 2 = net radiation energy exchanged between 1 and 2 F1− 2 = fraction of radiation energy leaving 1 and reaching 2 F2−1 = fraction of radiation energy leaving 2 and reaching 1 25

For black surfaces: Radiation that leaves the surface 1:

Q&1 = S1Eb1

and is intercepted by the surface 2:

F1− 2 S1Eb1

Radiation that leaves the surface 2:

Q& 2 = S 2 Eb 2

and is intercepted by the surface 1:

F2−1S 2 Eb 2

The net energy exchanged between the surfaces 1 and 2:

Q& 1- 2 = S1F1- 2 Eb1 − S2 F2-1 Eb 2

(a)

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If T1 = T2 then E b1 = E b 2 and Q&12 = 0.

S1F1− 2 = S2 F2−1

(b)

Reciprocal rule (vztah recoprocity) Combine (a) and (b) and use Stefan-Boltzmann law

Eb = σ T 4 :

(

Q& 1- 2 = S1F1- 2 ( Eb1 − Eb 2 ) = S1F1- 2σ T14 − T24

)

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(2) Rules: • Reciprocal rule can be generalized S i Fi − j = S j F j − i

• Additive rule: Conservation of energy - see the figure. F1-(2+ 3 ) = F1- 2 + F1- 3

2

3

Multiply by S1 S1F1−( 2+ 3) = S1F1- 2 + S1F1- 3

Use the reciprocal rule

1 Fig. 10.5

( S 2 + S 3 )F(2+ 3)-1 = S 2 F2-1 + S 3 F3-1

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• Enclosure or summation rule: All energy leaving one surface must be received by some or all other surfaces

F11 + F12 + F13 + K + F1n = 1 n

∑ Fij = 1

i = 1,2,3, K , n

j =1

3

2

4

1

n Fig. 10.6

• Conclusion: Fii = 0 for a plane or convex surface and Fii ≠ 0 for a concave surface (3) Determination of view factors: • Simple configurations: By physical reasoning: 29

F12 = 1 Apply the reciprocal rule

S 2 F2−1 = S1F1− 2

A1 A2 1

2

F2−1 = ( S1 / S2 ) F1− 2 = S1 / S2 • Other methods: • Surface integration method: Can involve tedious double integrals • View factor algebra method: Known factors are used in a superposition scheme together with the three view factor rules to construct factors for other configurations 30

View factor for parallel rectangles 31

View factor for perpendicular rectangles with a common side 32